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Transmutation theory and rank for quantum braided groups

Published online by Cambridge University Press:  24 October 2008

Shahn Majid
Shahn Majid
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 9EW
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Abstract

Let f: H1 → H2be any pair of quasitriangular Hopf algebras over k with a Hopf algebra map f between them. We construct in this situation a quasitriangular Hopf algebra B(H1, f, H2) in the braided monoidal category of H1-modules. It consists in the same algebra as H2 with a modified comultiplication and has a quasitriangular structure given by the ratio of those of H1 and H2. This transmutation procedure trades a non-cocommutative Hopf algebra in the category of k-modules for a more cocommutative object in a more non-commutative category. As an application, every Hopf algebra containing the group algebra of ℤ2 becomes transmuted to a super-Hopf algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 113 , Issue 1 , January 1993 , pp. 45 - 70
Copyright
Copyright © Cambridge Philosophical Society 1993

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References

REFERENCES

[1]Deligne, P. and Milne, J. S.. Tannakian Categories. Lecture Notes in Math. vol. 900 (Springer-Verlag, 1982).Google Scholar
[2]Drinfeld, V. G.. Quantum groups. In Proceedings of the International Congress of Mathematicians, Berkeley 1986 (American Mathematical Society, 1987), pp. 798–820.Google Scholar
[3]Freyd, P. and Yetter, D.. Braided compact closed categories with applications to low dimensional topology. Adv. in Math. 77 (1989), 156–182.CrossRefGoogle Scholar
[4]Gurevich, D. I. and Majid, S.. Braided groups of Hopf algebras obtained by twisting. Pacific J. Math., to appear.Google Scholar
[5]Jimbo, M.. A q-difference analog of U(g) and the Yang–Baxter equation. Lett. Math. Phys. 10 (1985), 63–69.CrossRefGoogle Scholar
[6]Joyal, A. and Street, R.. Braided monoidal categories. Mathematics Reports 86008, Macquarie University (1986).Google Scholar
[7]Joyal, A. and Street, R.. The geometry of tensor calculus, I. Adv. in Math. 8 (1991), 55–112.CrossRefGoogle Scholar
[8]Lyubashenko, V. V. and Majid, S.. Braided groups and quantum fourier transform. Preprint (1991).Google Scholar
[9]Macfarlane, A. and Majid, S.. Quantum group structure in a fermionic extension of the quantum harmonic oscillator. Phys. Lett. B 268 (1991), 71–74.CrossRefGoogle Scholar
[10]MacLane, S.. Categories for the Working Mathematician. Graduate Texts in Math. vol. 5 (Springer-Verlag, 1974).Google Scholar
[11]Majid, S.. Quasi-triangular Hopf algebras and Yang–Baxter equations. Internal. J. Modern Phys. A 5 (1990), 1–91.CrossRefGoogle Scholar
[12]Majid, S.. Physics for algebraists: Non-commutative and non-cocommutative Hopf algebras by a bicrossproduct construction. J. Algebra 130 (1990), 17–64.CrossRefGoogle Scholar
[13]Majid, S.. Braided groups and algebraic quantum field theories. Lett. Math. Phys. 22 (1991), 167–176.CrossRefGoogle Scholar
[14]Majid, S.. Anyonic quantum groups. Preprint (1991).Google Scholar
[15]Majid, S.. Examples of braided groups and braided matrices. J. Math. Phys. 32 (1991). 3246–3253.CrossRefGoogle Scholar
[16]Majid, S.. Cross products by braided groups and bosinization. J. Algebra, to appear.Google Scholar
[17]Majid, S.. Principle of representation-theoretic self-duality. Phys. Essays 4 (1991), 395–405.CrossRefGoogle Scholar
[18]Majid, S.. Reconstruction theorems and rational conformal field theories. Internal. J. Mod. Phys. 6 (1991), 4359–4374.CrossRefGoogle Scholar
[19]Majid, S.. Braided groups. J. Pure Appl. Algebra, to appear.Google Scholar
[20]Majid, S.. Representations, duals and quantum doubles of monoidal categories. Rend. Circ. Mat. Palermo (2) Suppl. 26 (1991), 197–206.Google Scholar
[21]Majid, S.. Representation-theoretic rank and double Hopf algebras. Comm. Algebra 18 (11): 3705–3712 (1990), 3705–3712.CrossRefGoogle Scholar
[22]Majid, S.. Doubles of quasitriangular Hopf algebras. Comm. Algebra 19 (1991), 3061–3073.CrossRefGoogle Scholar
[23]Manin, Yu. I.. Quantum groups and non-commutative geometry. Technical report, Centre de Recherches Math, Montreal (1988).Google Scholar
[24]Pareigis, B.. A non-commutative non-cocommutative Hopf algebra in nature. J. Algebra 70 (1981), 356–374.CrossRefGoogle Scholar
[25]Radford, D.. On the quasitriangular structures of a semisimple Hopf algebra. J. Algebra 141 (1991), 354–358.CrossRefGoogle Scholar
[26]Reshetikhin, N. Yu. and Turaev, V. G.. Ribbon graphs and their invariants derived from quantum groups. Comm. Math. Phys. 127 (1990), 1–26.CrossRefGoogle Scholar
[27]Rivano, N. Saavedra. Catégories Tannakiennes. Lecture Notes in Math. vol. 265 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[28]Sweedler, M. E.. Hopf Algebras (Benjamin, 1969).Google Scholar
[29]Ulbrich, K.-H.. On Hopf algebras and rigid monoidal categories. Israel J. Math. 72 (1990), 252–256.CrossRefGoogle Scholar